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\begin{document}
\title{Investment in mining and new energy technologies}
\pagenumbering{arabic}
\maketitle
\date{}
We model economic activity in continuous time, indexed by $t$. The state variables, the controls, and the technology variables are functions of time. We shall usually simplify notation, however, by omitting time as an explicit argument.

\section{Goods and services production}

There is a single consumption good in the economy. Letting $c$ denote per capita consumption, we assume that the lifetime
utility function is given by: 
\begin{equation}
U=\max \int_{0}^{\infty }e^{-\beta \tau }\frac{c(\tau )^{1-\gamma }}{%
1-\gamma }\,d\tau  \label{eq:Objective}
\end{equation}
The term $e^{-\beta \tau }$ acts as a discount factor, capturing the fact
that utility from future consumption is less valuable than today's
consumption.

The production function is an $Ak$ model augmented by an explicit accounting for energy input. Specifically, for a per capita capital stock of $k$ producing per capita output of goods and services of $y$, we assume per capita energy input $e=Fk$ is required where $F$ can be interpreted as the fuel intensity of the capital stock.\footnote{The usual energy identity relates energy input $e$ to delivered energy service $u$ per unit of capital, such as miles per vehicle, times the capital stock $k$, such as the number of vehicles, divided by energy efficiency $\epsilon$, such as miles per gallon. In the vehicle example this yields total gallons of fuel consumed. The fuel intensity $F$ then is $u/\epsilon$.} We will allow for investments $i_F$ to lower fuel intensity below its initial value $F_0$ by raising energy efficiency until $F$ attains a technological lower limit $\bar{F}$. Specifically, we define $f$ as the accumulated investment in efficiency improvements so
\begin{equation}
\dot{f}=i_F
\label{eq:EffGain}
\end{equation}
and for simplicity we assume a linear functional form
\begin{equation}
F=
\begin{cases}
F_0-af & \text{ if $f\le (F_0-\bar{F})/a$},
\\ \bar{F} & \text{ otherwise}
\end{cases}
\label{eq:Intensity}
\end{equation}
where $a>0$ and by definition of $F_0$ as the initial value of $F$ we take $f(0)=0$. 

Energy can be provided by two different technologies that also require capital investments to produce useful output.  One, with capital stock denoted $k_R$ mines fossil fuels that are depletable and converts them into useable energy products. The other, with capital stock denoted $k_B$, is a backstop or renewable technology where the energy source itself is ``harvested'' from the environment using the capital equipment, so there is no resource depletion, although there are operating and maintenance costs. Since the energy source for the backstop technology is in unlimited supply, once the economy switches to the backstop technology the economy becomes an endogenous growth model with perpetual per capita economic growth. The two types of energy are perfect substitutes for producing goods output, but once energy-producing capital is in place it cannot be converted from one type to the other. 

Each type of capital (the goods-producing capital and the two-energy producing capital stocks) is accumulated via investment $i, i_B, i_R$ and depreciates at the rate $\delta$:\footnote{Although different types of capital could depreciate at different rates, the data we use to calibrate the model provides only a single rate of depreciation for capital.}
\begin{equation}
\dot{k}=i-\delta k
\label{eq:kdot}
\end{equation}
\begin{equation}
\dot{k}_B=i_B-\delta k_B
\label{eq:kBdot}
\end{equation}
\begin{equation}
\dot{k}_R=i_R-\delta k_R
\label{eq:kRdot}
\end{equation}

Total energy input into goods production is given by $e=R+B$ where $R$ equals the energy produced using $k_R$ and $B$ the energy produced using $k_B$.  We also assume linear production functions for the energy producing industries
\begin{gather}
R=\mu_Rk_R \notag
\\ B=\mu_Bk_B
\label{eq:EnergyProd}
\end{gather}

We assume that, in order to produce per capita fossil energy output $R$, primary fossil fuel inputs are required.  We assume that the marginal costs of resource extraction plus conversion increase with the total quantity of resources mined to date, $S$.\footnote{Heal (1976) introduced the idea of an increasing marginal cost of extraction to show that the optimal price of an exhaustible resource begins above marginal cost, and falls toward it over time. This claim is rigorously proved in Oren and Powell (1985).} Letting $Q$ denote the (exogenous) population and labor supply, the total fossil fuel used will be $\rho QR$, and then $S$ will satisfy:
\begin{equation}
\dot{S}=\rho QR
\label{eq:Sdot}
\end{equation}
We will assume that $Q$ grows at the constant rate $\pi$.

We further modify the resource depletion model to allow for technical change in mining and conversion. Explicitly, we assume that the per unit cost of mining and conversion, $g(S,N)$, depends not only on $S$ but also the state of technical knowledge $N$, which can be augmented through investment: 
\begin{equation}
\dot{N}=n 
\label{eq:Ndot}
\end{equation}
Per capita mining and conversion costs will then be given by $g(S,N)\rho R=g(S,N)\rho \mu_Rk_R$. We can interpret $n$ as investments that raise the productivity of capital in the mining and conversion industries. This could include new new discoveries, improvements in mining technology and improvements in conversion efficiency. The latter reduces the primary energy input required to supply a given amount of useful secondary energy.

We assume that $g(S,N)$ is given by the following function: 
\begin{equation}
g(S,N)=\alpha _{0}+\frac{\alpha _{1}}{\bar{S}-S-\alpha _{2}/(\alpha_{3}+N)}=\alpha _{0}+\frac{\alpha _{1}(\alpha _{3}+N)}{(\bar{S}-S)(\alpha_{3}+N)-\alpha _{2}}  \label{eq:MCMining}
\end{equation}%
illustrated in Figure \ref{fig:MiningMC}. For a given state of technical knowledge $N$, mining and conversion costs become unbounded as $S\to \bar{S}-\alpha _{2}/(\alpha _{3}+N)$. The absolute maximum fossil fuel available is given by $\bar{S}$, and this is only available asymptotically as the stock of investment in new fossil fuel technology $N\to\infty$. The terms $\alpha_{0},\alpha _{1},\alpha _{2}$ and $\alpha _{3}$ in \eqref{eq:MCMining} are parameters. We define $N$ and $S$ so that initially $N(0)=S(0)=0$.

\begin{figure}[ht]
\centering \includegraphics[width=3.5in]{MiningMC.pdf}
\caption{Marginal cost of mining fossil fuels}
\label{fig:MiningMC}
\end{figure}

Since $g$  increases without bound as new investments $N$ can no longer compensate for the effects of depletion $S$, we have implicitly assumed that depletion will ultimately swamp improvements in mining technology and conversion efficiency. A consequence of this assumption is that fossil fuel use will be discontinued at some point. As noted above, for mining to cease at that time, the utilization $\rho$ of $k_R$ has to fall to zero and remain at zero thereafter. Also, once fossil fuel use ceases,  $S, N$ and $g$ will remain constant.

For the later analysis, it also is useful to derive the partial derivatives of the fossil fuel cost function $g(S,N)$. The fist partial derivatives are given by 
\begin{equation}
\frac{\partial g}{\partial S}=\frac{\alpha_1(\alpha_3+N)^2}{[(\bar{S}-S)(\alpha_3+N)-\alpha_2]^2}>0 
\label{eq:PartialgPartialS}
\end{equation}
and 
\begin{equation}
\frac{\partial g}{\partial N}=-\frac{\alpha_1\alpha_2}{[(\bar{S}-S)(\alpha_3+N)-\alpha_2]^2}<0 
\label{eq:PartialgPartialN}
\end{equation}
so that increases in $S$ increase marginal cost, while improved technology reduces the costs of providing fossil fuel energy. The second order partial derivatives with respect to $S$ and $N$ are given by 
\begin{equation}
\frac{\partial^2 g}{\partial S^2}=\frac{2\alpha_1(\alpha_3+N)^3}{[(\bar{S}-S)(\alpha_3+N)-\alpha_2]^3}>0  \label{eq:Partial2gPartialS2}
\end{equation}
and 
\begin{equation}
\frac{\partial^2 g}{\partial N^2}=\frac{2\alpha_1\alpha_2(\bar{S}-S)}{[(\bar{S}-S)(\alpha_3+N)-\alpha_2]^3}>0  \label{eq:Partial2gPartialN2}
\end{equation}
In particular, this function implies that cumulative exploitation $S$ increases fossil fuel energy cost at an increasing rate, while investment in fossil fuel technology decreases costs at a decreasing rate. In fact, we can conclude from \eqref{eq:PartialgPartialN} that $\partial g/\partial N\rightarrow 0$ as $N\rightarrow\infty$. The latter fact should imply that eventually it becomes uneconomic to invest further in reducing the costs of fossil fuel energy. Thus, fossil fuel resources will likely be abandoned long before all known deposits are exhausted as rising costs make renewable energy technologies more attractive.
Finally, the cross second partial derivative will be given by 
\begin{equation}
\frac{\partial^2 g}{\partial N\partial S}=-\frac{2\alpha_1\alpha_2(%
\alpha_3+N)}{[(\bar{S}-S)(\alpha_3+N)-\alpha_2]^3}<0
\label{eq:Partial2gPartialSN}
\end{equation}
Hence, investment in fossil fuel technology delays the increase in costs of fossil fuel energy accompanying increased exploitation.

With regard to renewable technologies, we assume that operating and maintenance costs per unit of installed energy-producing capital $k_B$ are given by $m$. In addition, of course, the production of energy from renewable sources requires investment in $k_B$. We allow technological progress to increase $\mu_B$, and hence reduce the amount of capital $k_B$ required to yield a given level of energy output $B$. Explicitly, using $h$ to denote the stock of knowledge about backstop energy production, we assume:
\begin{equation}
\mu_B=
\begin{cases}
H_0+bh & \text{ if $h\le (\bar{H}-H_0)/b$},
\\
\bar{H} & \text{ otherwise}
\end{cases}
\label{eq:RenewCost}
\end{equation}
where $b>0$ is a constant, $H_0$ is the initial value of $\mu_B$, so $h(0)=0$, and $\bar{H}$ is an upper limit, determined by physical constraints, above which $\mu_B$ cannot rise.

Following the learning by doing literature, we assume that experience constructing capital increases the stock of knowledge $h$. However, we also allow direct R\&D expenditure $j$ to accelerate the accumulation of knowledge about the renewable technology.\footnote{Klaassen et. al. (2005) studied the impact of public R\&D and capacity expansion on cost reducing innovation for wind turbine farms in Denmark, Germany and the UK. They estimated a two-factor learning curve model that allowed for both learning-by-doing and direct R\&D. They derive robust estimates suggesting that direct R\&D is roughly twice as productive for reducing costs as is learning-by-doing. They interpret their results as enhancing the validity of the two-factor learning curve formulation. Kouvaritakis et al. (2000) used a two-factor learning specification that incorporates learning-by-doing effects as well as a relationship between technology performance and R\&D expenditure.} Once $h$ reaches its upper limit, further investment in the technology would be worthless and we should have $\dot{h}=0$. The parameter $\psi$
determines the contribution from  experience to the stock of knowledge $h$ 
\begin{equation}
\dot{h}=
\begin{cases}
k_B^\psi j^{1-\psi} & \text{ if $h\leq (\bar{H}-H_0)/b$}, \\ 
0 & \text{ otherwise}
\end{cases}
\label{eq:Hdot}
\end{equation}
In particular, once $h$ reaches its upper limit, further investment in the technology would be worthless and we should have $j=0$. The parameter $\psi$
determines how explicit investment in research enhances the accumulation of knowledge from experience.

Goods output is consumed, used to produce energy, or invested in $k, k_R, k_B, f, N$ or $h$. Let $c$ denote per capita consumption. This leads to a budget constraint (in per capita terms):
\begin{equation}
 c+i+i_R+i_B+i_F+n+j+g(S,N)\rho \mu_Rk_R+mk_B=Ak
\label{eq:Budget}
\end{equation}

Also, equilibrium in the energy market requires
\begin{equation}
Fk=\rho\mu_Rk_R+\mu_Bk_B
\label{eq:EnEquil}
\end{equation}
where $0\le \rho \le 1$ is a utilization variable. In particular, note that since capital depreciates exponentially a positive amount of $k_R$ will remain even after fossil fuel becomes too expensive and investment in $k_R$ ceases. Further mining and conversion costs can then be avoided by choosing $\rho=0$ and ceasing to use $k_R$ to provide energy. Since increased output of energy from a given $k_B$ does not incur additional costs, any amount of $k_B$ that is available will always be fully utilized.\footnote{This corresponds in practice, for example, to using all wind, solar, geothermal or run-of-river hydro output that is produced. Formally, it can be shown that if we introduce a variable $0\le \rho_B \le 1$ analogous to $\rho$ we would always have $\rho_B=1$ when $k_B>0$.}

Observe that we can use the right side of \eqref{eq:EnEquil} to write $Ak$ in terms of $k_R$ and $k_B$. After doing so, the budget constraint \eqref{eq:Budget} can be written in terms of the \textit{net} contribution to output from the energy sector:
\begin{equation}
c+i+i_R+i_B+i_F+n+j=\rho\mu_Rk_R\bigl [\frac{A}{F}-g(S,N)\bigr ] + \mu_Bk_B\bigl (\frac{A}{F}-m\bigr )
\label{eq:NetBudget}
\end{equation}
We assume that the productivity $A$ of capital $k$ in producing output is large enough, and the initial fuel intensity $F_0$ is low enough, that $A/F_0$ exceeds both $g(0,0)$ and $m$.

\section{The general optimization problem}

The objective function \eqref{eq:Objective} is maximized subject to the differential constraints \eqref{eq:EffGain}, \eqref{eq:kdot}, \eqref{eq:kBdot}, \eqref{eq:kRdot}, \eqref{eq:Sdot}, \eqref{eq:Ndot} and \eqref{eq:Hdot} and with initial conditions $k(0)=k_0, k_R=k_{R0}$ and $f(0)=h(0)=k_B(0)=S(0)=N(0)=H(0)=0$, the budget constraint \eqref{eq:Budget}, and the energy market equilibrium condition \eqref{eq:EnEquil}. The control variables are $c, \rho, i, i_{R}, i_{B}, i_F, n$ and $j$, while the state variables are $k, k_{R}, k_{B}, f, N, S$ and $h$. Denote the corresponding co-state variables by  $q, q_{R}, q_{B}, \varphi, \nu, \sigma$ and $\eta$.  Let $\lambda$ be the Lagrange multiplier on the budget constraint and $p_e$ the multiplier on the energy market equilibrium constraint.

We also need to allow for the possibility that either $k_R$ or $k_B$ is not used, while the utilization rate $\rho$ is by definition bounded above by 1. To that end, we use $\theta_{L}$ and $ \theta_{U}$ for the Lagrange multipliers on the inequality constraints on $\rho$. Also use $\omega_F$ to denote the multiplier on the constraint $i_F\ge 0$, $\omega_N$ to denote the multiplier on the constraint $n\ge 0$, $\omega_{H}$ the multiplier on the constraint $j\ge 0$, $\omega_{R}$ the multiplier on the constraint $i_{R}\ge 0$ and $\omega_{B}$ the multiplier on the constraint $i_{B}\ge 0$.

For the situation where both $f$ and $h$ are below their upper bounds, define the current value Hamiltonian and Lagrangian by
\begin{equation}
\begin{split}
\mathcal{H} &=\frac{c^{1-\gamma}}{1-\gamma}+q(i-\delta k)+q_{R}(i_{R}-\delta k_{R})+q_{B}(i_{B}-\delta k_{B})+\varphi i_F+\nu n+\sigma\rho Q\mu_Rk_R+
\\&\eta k_B^\psi j^{1-\psi}+\lambda\biggl\{Ak-c-i-i_{R}-i_{B}-i_F-n-j-g(S,N)\rho \mu_Rk_R-mk_B\biggr\}+
\\&p_e\bigl \{\rho\mu_Rk_R+(H_0+bh)k_B-(F_0-af)k\bigr \}+\theta_{L}\rho+\theta_{U}(1-\rho)+\omega_Fi_F+\omega_{N} n+\\&\omega_{H}j+\omega_{R}i_{R}+\omega_{B}i_{B}
\end{split}
\label{eq:Hamiltonian}
\end{equation}

The first order conditions for a maximum with respect to the control variables are:
\begin{equation}
\frac{\partial \mathcal{H}}{\partial c}= c^{-\gamma} -\lambda = 0 \label{eq:FOCc}
\end{equation}
\begin{equation}
\begin{split}
&\frac{\partial \mathcal{H}}{\partial \rho}= \sigma Q\mu_Rk_R-\lambda\mu_Rk_Rg(S,N)+p_e\mu_Rk_R+\theta_{L}-\theta_{U} = 0 
\\ \theta_{L}\rho&=0,  \theta_{L}\ge 0, \rho\ge 0, \theta_{U}(1-\rho)=0, \theta_{U}\ge 0, \rho\le 1
\end{split}
\label{eq:FOCrho}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial i}=q-\lambda=0
\label{eq:FOCi}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial i_{R}}=q_{R} -\lambda + \omega_{R} = 0;  \omega_{R}i_{R}=0, \omega_{R}\ge 0, i_{R}\ge 0
\label{eq:FOCiR}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial i_{B}}=q_{B} -\lambda + \omega_{B} = 0;  \omega_{B}i_{B}=0, \omega_{B}\ge 0, i_{B}\ge 0
\label{eq:FOCiB}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial i_F}=\varphi -\lambda + \omega_F = 0;  \omega_Fi_F=0, \omega_F\ge 0, i_F\ge 0
\label{eq:FOCiF}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial n}= \nu-\lambda +\omega_N = 0, \omega_N n=0, \omega_N\ge 0, n\ge 0 \label{eq:FOCn}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial j}= \eta(1-\psi)k_B^\psi j^{-\psi}-\lambda +\omega_H = 0, \omega_H j=0, \omega_H\ge 0, j\ge 0
\label{eq:FOCj}
\end{equation}

The differential equations for the co-state variables are:
\begin{equation}
\dot{q}=\beta q-\frac{\partial \mathcal{H}}{\partial k}=(\beta+\delta)q -\lambda A + p_e(F_0-af)
\label{eq:qdot}
\end{equation}\begin{equation}
\dot{q}_R = \beta q_R-\frac{\partial \mathcal{H}}{\partial k_R} = (\beta+\delta) q_R-\sigma\rho Q\mu_R+\lambda g(S,N)\rho\mu_R-p_e\rho\mu_R  \label{eq:qRdot}
\end{equation}
\begin{equation}
\dot{q}_B = \beta q_B-\frac{\partial \mathcal{H}}{\partial k_B} = (\beta+\delta) q_B-\eta\psi k_B^{\psi-1}j^{1-\psi}+\lambda m-p_e(H_0+bh)
\label{eq:qBdot}
\end{equation}
\begin{equation}
\dot{\varphi}=\beta\varphi-\frac{\partial \mathcal{H}}{\partial f}=\beta\varphi -p_eak
\label{eq:varphidot}
\end{equation}
\begin{equation}
\dot{\nu} = \beta \nu - \frac{\partial \mathcal{H}}{\partial N} = \beta\nu +\lambda\rho\mu_{R}k_R\frac{\partial g}{\partial N} 
\label{eq:nudot}
\end{equation}
\begin{equation}
\dot{\sigma} = \beta \sigma - \frac{\partial \mathcal{H}}{\partial S} = \beta \sigma +\lambda\rho\mu_{R}k_R\frac{\partial g}{\partial S} 
\label{eq:sigmadot}
\end{equation}
\begin{equation}
\dot{\eta} = \beta \eta- \frac{\partial \mathcal{H}}{\partial h} =\beta\eta -p_ebk_B
\label{eq:etadot}
\end{equation}
We also recover the budget constraint \eqref{eq:Budget}, the energy market equilibirum condition \eqref{eq:EnEquil}, and the differential equations for the state variables, \eqref{eq:EffGain}, \eqref{eq:kdot}, \eqref{eq:kBdot}, \eqref{eq:kRdot},  \eqref{eq:Sdot}, \eqref{eq:Ndot} and \eqref{eq:Hdot}.

Let $V$ denote the maximized value of the objective function \eqref{eq:Objective} subject to the constraints. Then from the Hamilton-Jacobi-Bellman equation, we have $-V_t=\max e^{-\delta t}\mathcal{H}$ while the co-state variables satisfy $q=\partial V/\partial k, q_R=\partial V/\partial k_R, q_B=\partial V/\partial k_B, \varphi= \partial V/\partial f, \sigma=\partial V/\partial S, \nu=\partial V/\partial N$ and $\eta=\partial V/\partial h$.

Suppose fossil fuel ceases to be used at $T_5$, so $\rho(t)=0$ for $t>T_5$.\footnote{The subscript arises since we later define other critical times $T_i, 1=1, 2, 3, 4$ which occur earlier than $T_5$.}  Since further changes in $S$ or $N$ then have no effect on $V$, $\sigma=\nu=0$ at $T_5$. From \eqref{eq:FOCrho}, the shadow price of energy $p_e=\lambda g$ at $T_5$. At the time fossil fuel is abandoned, the energy price just equals the marginal cost of energy production in terms of goods. Prior to $T_5$, the shadow energy price includes the user cost of fossil fuel production $-\sigma Q$, noting that $\sigma$, the shadow price of $S$, is negative.

Before returning to analyze the economy with fossil energy, in the next three sections of the paper we analyze the simpler regimes that use only renewable energy. We then focus on the initial economy, which we assume the economy uses only fossil fuels. Finally, we will examine the intermediate regimes where both energy sources are used. The analysis in these intermediate regimes combines elements from the regimes where only one energy source is used.

\subsection{The long run endogenous growth economy}

In the very long run, the possibilities for reducing the fuel intensity of goods production or increasing the efficiency of renewable energy production are exhausted. The variables $F$ and $\mu_B$ become constants. The choice variables decrease in number to just $c, i$ and $i_B$, while the state variables are just $k$ and $k_B$. In this regime, the budget constraint simplifies to
\begin{equation}
c+i+i_B+mk_B=Ak
\label{eq:Termbudget}
\end{equation}
while the energy market equilibrium condition becomes
\begin{equation}
\bar{F}k=\bar{H}k_B
\label{eq:TermEnEquil}
\end{equation}
Equation \eqref{eq:TermEnEquil} implies that the two capital stocks must be maintained in a constant ratio. Since the marginal costs of investing in the two types of capital are the same, the marginal benefits also must be kept the same and the economy moves along a balanced growth path. To maintain this balanced growth path, the two investments cannot be chosen separately. Differentiating \eqref{eq:TermEnEquil} and using \eqref{eq:TermEnEquil} and the assumption that the depreciation rates are identical, we also obtain
\begin{equation}
\bar{F}i=\bar{H}i_B
\label{eq:TermInvestments}
\end{equation}
Using \eqref{eq:TermEnEquil} and \eqref{eq:TermInvestments}, the budget constraint can then be simplified to
\begin{equation}
c+\bigl (1+\frac{\bar{F}}{\bar{H}}\bigr )i = (A-\frac{m\bar{F}}{\bar{H}})k
\label{eq:SimpBudget}
\end{equation}

We now simply want to choose $c$ and $i$ to maximize
\begin{equation}
\tilde{\mathcal{H}}=\frac{c^{1-\gamma}}{1-\gamma}+q(i-\delta k)+\lambda\biggl [(A-\frac{m\bar{F}}{\bar{H}})k-c-\bigl (1+\frac{\bar{F}}{\bar{H}}\bigr )i \biggr ]
\label{eq:TermHamil}
\end{equation}
The first order conditions for the choice of the controls become
\begin{equation}
c^{-\gamma}=\lambda
\label{eq:TermFOCc}
\end{equation}
and
\begin{equation}
q=\lambda\bigl (1+\frac{\bar{F}}{\bar{H}}\bigr )
\label{eq:TermFOCi}
\end{equation}
while the co-state equation for $q$ becomes
\begin{equation}
\dot{q}=(\beta+\delta)q-\lambda(A-\frac{m\bar{F}}{\bar{H}})
\label{eq:Termqdot}
\end{equation}
Differentiating \eqref{eq:TermFOCi} and substituting the result into \eqref{eq:Termqdot} we obtain
\begin{equation}
\frac{\dot{\lambda}}{\lambda}=\beta+\delta-\frac{A\bar{H}-m\bar{F}}{\bar{H}+\bar{F}}\equiv -\bar{A}
\label{eq:Termlamdot}
\end{equation}
where $\bar{A}$ is a constant. If we are to have perpetual growth, we must have $c\to\infty$ as $t\to\infty$, which from \eqref{eq:TermFOCc} will require $\lambda\to 0$ and hence $\bar{A}>0$, that is
\begin{equation}
A>\beta+\delta+\frac{\bar{F}}{\bar{H}}(\beta+\delta+m)
\label{eq:ACond1}
\end{equation}
High productivity $A$ of $k$ in the production of goods, a low terminal fuel intensity $\bar{F}$ of goods production, a high terminal efficiency $\bar{H}$ of renewable energy production, and a low discount rate $\beta$, depreciation rate $\delta$ and operating and maintenance cost of renewable energy production $m$ all make it more likely that \eqref{eq:ACond1} will hold. Hereafter, we assume \eqref{eq:ACond1} to be valid.

The solution to \eqref{eq:Termlamdot} can be written
\begin{equation}
\lambda=\bar{K}e^{-\bar{A}t}
\label{eq:term_lam_sol}
\end{equation}
for some constant $\bar{K}$ yet to be determined. Thus, in this final regime, the differential equation for $k$, the first order condition \eqref{eq:TermFOCc} for $c$ and the budget constraint \eqref{eq:SimpBudget} imply
\begin{equation}
\dot{k}=(\bar{A}+\beta)k-\frac{\bar{H}\bar{K}^{-1/\gamma}e^{\bar{A}t/\gamma}}{\bar{H}+\bar{F}}
\label{eq:TermBudget}
\end{equation}
The integrating factor for the differential equation \eqref{eq:TermBudget} is $e^{-(\bar{A}+\beta)t}$, so the solution can be written
\begin{equation}
k = C_0e^{(\bar{A}+\beta)t}+\frac{\bar{H}\gamma \bar{K}^{-1/\gamma}e^{\bar{A}t/\gamma}}{(\bar{H}+\bar{F})[\beta\gamma+\bar{A}(\gamma-1)]}
\label{eq:Term_k_gensol}
\end{equation}
for another constant $C_0$. However, the transversality condition requires
\begin{equation}
\lim_{t\to\infty}e^{-\beta t}\lambda k = C_0\bar{K}+\lim_{t\to\infty}\frac{\bar{H}\gamma \bar{K}^{(1-1/\gamma)}e^{(\bar{A}/\gamma-\bar{A}-\beta)t}}{(\bar{H}+\bar{F})[\beta\gamma+\bar{A}(\gamma-1)]}=0
\label{eq:TVC}
\end{equation}
Equation \eqref{eq:TVC} in turn requires
\begin{equation}
C_0=0\,\text{  and  }\,\bar{A}(1-\gamma)<\beta\gamma
\label{eq:TVCresult}
\end{equation}
Note that since $\bar{A}>0$ the inequality in \eqref{eq:TVCresult} will be satisfied if $\gamma>1$,  while if $0<\gamma<1$, it will require
\begin{equation}
A<\frac{\beta}{1-\gamma}+\delta+\frac{\bar{F}}{\bar{H}}(\frac{\beta}{1-\gamma}+\delta+m)
\label{eq:ACond2}
\end{equation}
which further limits the range of acceptable parameter values relative to \eqref{eq:ACond1}.

In summary, we conclude that the value of $k$ in the final endogenous growth economy will be given by
\begin{equation}
k = \frac{\bar{H}\gamma \bar{K}^{-1/\gamma}e^{\bar{A}t/\gamma}}{(\bar{H}+\bar{F})[\beta\gamma+\bar{A}(\gamma-1)]}
\label{eq:Term_k_sol}
\end{equation}
with $\lambda$ given by \eqref{eq:term_lam_sol} and where $\bar{K}$ is a constant yet to be determined. From \eqref{eq:TermEnEquil} and \eqref{eq:Term_k_sol}, the capital stock allocated to renewable energy production in the final endogenous growth economy will be
\begin{equation}
k_B = \frac{\bar{F}\gamma \bar{K}^{-1/\gamma}e^{\bar{A}t/\gamma}}{(\bar{H}+\bar{F})[\beta\gamma+\bar{A}(\gamma-1)]}
\label{eq:TermkBsol}
\end{equation}
The growth rate of the final economic regime will be
\begin{equation}
\frac{\bar{A}}{\gamma}=\frac{1}{\gamma}\biggl [\frac{A\bar{H}-m\bar{F}}{\bar{H}+\bar{F}}- (\beta+\delta)\biggr ]
\label{eq:AsymGr}
\end{equation}
In the usual endogenous growth model with a single capital stock $k$, linear production function $Ak$, depreciation rate of capital $\delta$, and representative consumer with time discount rate $\beta$ and constant relative risk aversion $\gamma$, the growth rate of the economy is $[A-(\beta+\delta)]/\gamma$. Comparing this with \eqref{eq:AsymGr}, we see that need for energy input to production reduces the \textit{net} productivity of capital $(A\bar{H}-m\bar{F})/(\bar{H}+\bar{F})$ by an amount that depends on the marginal cost of producing renewable energy $m$. Furthermore, the weights on gross capital productivity $A$ and marginal cost of energy production $m$ in net productivity depend on the relative values of the productivity of capital in producing renewable energy $\bar{H}$ and the energy intensity of final production $\bar{F}$.

\subsection{Discussion of the sequence of regimes}

The precise sequence of regimes prior to the terminal analytical one
just examined will depend on the parameter values. From previous
session, we know that investments in renewable energy $i_B$ and
end-use capital $i$ keep positive and continue beyond $\bar{T}$ into
the analytical regime. We also define $\hat{T}$ as the transition
period beyond which fossil fuels are no longer used. We will choose
parameter values so that the improvements in end-use energy efficiency
continue past $\hat{T}$.\footnote{A reason for expecting this sequence
  of events is that since reducing energy costs is only one objective
  in designing vehicles, buildings, appliances, equipment and so
  forth, energy efficiency may be compromised to enhance other
  features such as safety, visual attractiveness, increased comfort
  and so forth. Progress in improving end-use energy efficiency
  therefore tends to be slow. By contrast, since inefficiency in
  producing energy is a core cost for the energy producing sector, the
  drive to maximize energy efficiency is strong.} So that investments
in end-use fuel efficiency $i_F$ will be still positive after
$\hat{T}$. As long as both $i$ and $i_F$ are positive, the first order conditions \eqref{eq:FOCi} and \eqref{eq:FOCiF} imply $q=\lambda=\varphi$. Hence, $\dot{\lambda}=\dot{q}=\dot{\varphi}$ and from \eqref{eq:qdot} and \eqref{eq:varphidot} we conclude
\begin{equation}
\dot{\lambda}=(\beta+\delta)\lambda-\lambda A+p_e(F_0-af)=\beta\lambda-p_eak
\label{eq:iiFPos}
\end{equation}
Hence, $\forall t\le T_8$ the shadow price of energy $p_e$ is given by
\begin{equation}
p_e=\frac{\lambda(A-\delta)}{ak+F_0-af}
\label{eq:Enprice}
\end{equation}
Since the marginal costs of investing in end-use capital $k$ or
end-use energy efficiency $f$ are identical, the marginal benefits
also have to be the same. This effectively determines the price of
energy relative to end-use consumption goods.

We also assume, as an approximation,\footnote{Hydroelectricity
  provides more than half of primary energy that is non-fossil in
  origin. The next biggest source is nuclear. Remaining sources are
  very small and competitive at the moment primarily as a result of
  subsidies.} that initially fossil fuels alone are used. In other
words, while we assume that investments in end-use capital, fossil
fuel capital, mining and conversion technologies and end-use fuel
intensity initially are positive, that is, $i, i_R, n, i_F>0$ at
$t=0$, investments in renewable energy are zero, that is, $i_B=j=0$.

Let $T_1$ denote the time when regime 1 ends. This can, in principle,
result from any of four events. As already noted, we assume that
investments $i$ in end-use capital and $i_F$ in end-use fuel intensity
remain positive throughout the fossil regimes. However, regime 1 could
end because investment in mining $n$ or fossil fuel capital $i_R$
decline to zero at $T_1$, or investment $i_B$ in renewable energy
capital or $j$ in renewable energy efficiency become positive at
$T_1$. The most likely outcome, however, is that renewable energy
production and use will begin before investments in fossil fuel
capital or mining and conversion efficiency cease. Note that since we
have assumed $i_B=j=0$ in regime 1 and $k_B=H=0$ initially, we must
also have $k_B(T_1)=H(T_1)=0$.

We have assumed that explicit investment is an essential input to
improve the efficiency of renewable energy. Explicitly, the
differential equation of $h$, \eqref{eq:Hdot} implies that neither $j$
nor $k_B$ has effects unless they are both positive:

\begin{Prop}
  Before the technology progress $\mu_B$ reaches its upper limit
  $\bar{H}$, the investment in renewable energy efficiency $j = 0$ iff
  renewable energy capital $k_B = 0$.
\label{Prop:jzero1st}
\end{Prop}
\proof ``$\Rightarrow$'' From the first order condition
\eqref{eq:FOCj} for $j$, $\omega_H=\lambda-\eta\psi k_B=\lambda>0$ for
$k_B=0$ and hence $j=0$.

``$\Leftarrow$'' Given $j=0$,suppose we have $k_B>0$ instead of $k_B =
0$. Since $\mu_B<\bar{H}$, the shadow price of the cumulative
knowledge $\eta$, will keep positive. Then observed from \ref{eq:FOCj}
that we must also have $j^{\psi}\lambda\geq(1-\psi)\eta k_B^{\psi}>0$
and hence $j>0$. Contradiction with the assumption $j=0$. Hence
$k_B=0.$
\endproof
Also, when $j>0$, it must satisfy
\begin{align}
  \label{eq:j}
  j=\left(\frac{\lambda}{(1-\psi)\eta}\right)^{-1/\psi}k_B
\end{align}
As a result of Proposition~\ref{Prop:jzero1st}, we conclude that $k_B$
and $j$ as inputs will occur together. Therefore, we assume that
regime 1 will end as a result of both $i_B$ and $j$ becoming positive
while $i_R$ and $n$ remain positive.

% In fact, the proof of Proposition\ref{Prop:jzero1st} implies $j=0$
% for an interval $0\le k_B\le \lambda/\eta\psi$. Furthermore, a
% necessary condition for direct investment $j$ in $H$ to be positive
% is that the shadow value $\eta=\partial V/\partial H$ of $H$ is
% positive. In summary, we assume that regime 1 will end as a result
% of $i_B$ becoming positive while $i_R$ and $n$ remain positive.

After the transition point $\hat{T}$, we have $i_R = n = 0$, $i_F>0$
and $j>0$. And I find that $i_F$ will fall to zero first when $j$ is
still positive.
\begin{Prop}
  The regime $i_F>0$, $j=0$ cannot exist.
\end{Prop}
\begin{proof}
  Assume the regime $i_F>0$, $j=0$ exists. From the first order
  conditions, we find that $q=q_B=\varphi=\lambda$, and hence
  $\dot{q}=\dot{q_B}=\dot{\varphi}=\dot{\lambda}$. From
  $\dot{q_B}=\dot{\varphi}$ and equation \eqref{eq:Enprice}, we have
\begin{align}
  \label{eq:qBdot_vphidot_Hbar}
a(\delta+m)f-a(A+m)k = (\delta+m)F_0-(A-\delta)\bar{H}
\end{align}
from which, we obtain 
\begin{align}
  \label{eq:f_Hbar}
  f = \frac{A+m}{\delta+m}k-\frac{A-\delta}{a(\delta+m)}\bar{H}+\frac{F_0}{a}
\end{align}
Not that by definition, $0\leq f\leq
\frac{F_0-\bar{F}}{a}$. substituting \eqref{eq:f_Hbar} into it, we
have
\begin{align}
  \label{eq:krange_Hbar}
  k \in
  \left[\frac{(A-\delta)\bar{H}-(\delta+m)F_0}{a(A+m)},\frac{(A-\delta)\bar{H}-(\delta+m)\bar{F}}{a(A+m)}\right]
\end{align}
From $\dot{\lambda} = \dot{q_B}$ and \eqref{eq:Enprice} and \eqref{eq:f_Hbar}, we will have
\begin{align}
  \label{eq:lambdadot_qdot_Hbar}
\dot{\lambda} = \left[A-\frac{\delta\bar{H}+amk}{\bar{H}-ak}\right]k\lambda
\end{align}

where $A-\frac{\delta\bar{H}+amk}{\bar{H}-ak}<0$, so that
$k>\frac{(A-\delta)\bar{H}}{a(A+m)}$. From \eqref{eq:krange_Hbar}, we
will have $k<\frac{(A-\delta)\bar{H}}{a(A+m)}$, contradiction. Hence,
this regime does not exist.
\end{proof}


The sequence of regimes then can be illustrated as in
Figure~\ref{fig:Regimes}.
 
\begin{figure}[ht]
\centering \includegraphics[width=5.5in]{Regimes.pdf}
\caption{A schematic representation of the different regimes}
\label{fig:Regimes}
\end{figure}

We next analyze the solution for each of the regimes illustrated in
Figure~\ref{fig:Regimes}. As noted previously, we will begin with
renewable only regimes and then fossil fuel only regime, and regimes
where both kind of energy are used the last.

\subsection{Regime 6:  Investments in $i$, $i_B$, and renewable R\&D $j$}

In this regime, the end-use efficiency has already reached its frontier $F = \bar{F}$, while cumulative knowledge $H = H_0+bh<\bar{H}$. So the energy market equilibrium condition becomes
\begin{align}
  \label{eq:energyME_Fbar}
  \bar{F}k = (H_0+bh)k_B
\end{align}

% In this regime, the budget constraint becomes
% \begin{equation}
% c+i+i_F+i_B+mk_B=Ak
% \label{eq:Reg8Budget}
% \end{equation}
% while the energy market equilibrium condition becomes
% \begin{equation}
% (F_0-af)k=\bar{H}k_B
% \label{eq:Reg8EnEquil}
% \end{equation}
% In addition, we lose the differential equation \eqref{eq:Hdot} since $h$ is at its maximum and not changing. In addition, the corresponding co-state variable $\eta=0$. The first order conditions for the choice of the controls now imply
% \begin{equation}
% \lambda=q=q_B=\varphi
% \label{eq:Reg8FOCis}
% \end{equation}
% and we conclude from \eqref{eq:qBdot} and $\eta=0$ that, in addition to \eqref{eq:iiFPos}, $\dot{\lambda}$ satisfies
% \begin{equation}
% \dot{\lambda}=(\beta+\delta)\lambda+\lambda m-p_e\bar{H}
% \label{eq:Reg8Costate}
% \end{equation}
% Combining \eqref{eq:Reg8Costate} with \eqref{eq:iiFPos} and using \eqref{eq:Enprice} for $p_e$ we obtain a single equation linking $f$ and $k$:
% \begin{equation}
% a(\delta+m)f-a(A+m)k=(\delta + m)F_0-(A-\delta)\bar{H}
% \label{eq:Reg8_fk_rel}
% \end{equation}
% The energy market equilibrium condition \eqref{eq:Reg8EnEquil} and \eqref{eq:Reg8_fk_rel} then give two equations linking the three ``capital stocks'' (viewing $f$ as a form of intellectual as opposed to physical capital). Once again, we have a balanced growth path whereby investment has to keep the capital stocks in a definite relationship to each other.

% The need to maintain a balanced growth path imposes constraints on the investments $i, i_B$ and $i_F$. These can be derived by differentiating equations  \eqref{eq:Reg8EnEquil} and \eqref{eq:Reg8_fk_rel} and using the differential equations governing the evolution of $k, k_B$ and $f$. In particular, from \eqref{eq:Reg8_fk_rel} we find:
% \begin{equation}
% i_F=\frac{A+m}{\delta+m}(i-\delta k)
% \label{eq:Reg8iF}
% \end{equation}
% while \eqref{eq:Reg8EnEquil} together with \eqref{eq:Reg8iF} yields
% \begin{equation}
% i_B-\delta k_B=\frac{1}{\bar{H}}\biggl [F_0-af-\frac{A+m}{\delta+m}ak\biggr ](i-\delta k)
% \label{eq:Reg8iB}
% \end{equation}
% Substituting \eqref{eq:Reg8_fk_rel}, \eqref{eq:Reg8iF} and \eqref{eq:Reg8iB}, and the solution to \eqref{eq:FOCc} for $c$ into the budget constraint \eqref{eq:Reg8Budget}, one then obtains an equation that can be solved for investment $i$, namely:
% \begin{equation}
%  i = \frac{\bar{H}k(\delta+amk)}{2(\bar{H}-ak)}+\frac{\delta k}{2}-\frac{\bar{H}(\delta+m)\lambda^{-1/\gamma}}{2(A+m)(\bar{H}-ak)}
% \label{eq:Reg8i}
% \end{equation}
% Once we have a solution for $i$ in terms of $k$ and $\lambda$, \eqref{eq:Reg8iF} and \eqref{eq:Reg8iB} can be used to determine $i_F$ and $i_B$. The sole co-state variable in this regime, $\lambda$, will evolve according to \eqref{eq:Reg8Costate} with $p_e$ given by \eqref{eq:Enprice} and $f$ given by \eqref{eq:Reg8_fk_rel}:
% \begin{equation}
% \dot{\lambda} = \bigg(A-\frac{\delta\bar{H}+amk}{\bar{H}-ak}\bigg)k
% \label{eq:Reg8lamdot}
% \end{equation}
% The upper boundary of the regime, $T_8$, will occur where $f=(F_0-\bar{F})/a$. The values $k$ and $\lambda$ at $T_8$ must match the values of  $k$ and $\lambda$ at the beginning of the final analytical regime since these variables must be continuous across the boundary. As we noted in the previous section, however, the latter are only determined up to an unknown constant $\bar{K}$. In effect, we have two equations to determine  $\bar{K}$ and also the initial value of $\lambda$ at $t=0$. Finally, since $i_F>0$  throughout regime 8, the value of $\varphi$ at $T_8$ also has to equal the value of $\lambda$ at $T_8$.

% \subsection{Regime 6: Investment in $k$, $k_B$, renewable technology and end-use efficiency}

% The analysis in the previous section can be extended to the previous regime where both energy efficiency of end-use capital and the efficiency of renewable energy production are increasing, although explicit investment in R\&D to achieve the latter is absent. As the limit on the efficiency of renewable energy production looms, the incentive to invest in renewable R\&D will disappear since $h$ will continue to rise as a result of learning by doing even when $j=0$.

% The only differences relative to regime 8 are that the energy market equilibrium condition changes to
% \begin{equation}
% (F_0-af)k=(H_0+bh)k_B
% \label{eq:Reg7EnEquil}
% \end{equation}
% while $h$ evolves according to $\dot{h}=k_B$ and the corresponding co-state variable $\eta$ is no longer equal to zero. We thus obtain a new co-state equation:
% \begin{equation}
% \dot{\eta}=\beta\eta-p_ebk_B
% \label{eq:Reg7etadot}
% \end{equation}
% Also, the remaining co-state equation (apart from \eqref{eq:iiFPos}) that is analogous to \eqref{eq:Reg7Costate} becomes
% \begin{equation}
% \dot{\lambda}=(\beta+\delta)\lambda-\eta+\lambda m-p_e(H_0+bh)
% \label{eq:Reg7Costate}
% \end{equation}
% Paralleling the analysis in the previous section, \eqref{eq:Reg7Costate} can be combined with \eqref{eq:iiFPos} and expression \eqref{eq:Enprice} for $p_e$ to again yield a relationship between capital stocks that has to hold along the balanced growth path
% \begin{equation}
% (F_0-af)\biggl [\lambda(\delta+m)-\eta\biggr ]+ak\biggl [\lambda(A+m)-\eta\biggr ]=\lambda(A-\delta)(H_0+bh)
% \label{eq:Reg7fkrel}
% \end{equation}
% Once again, we can differentiate \eqref{eq:Reg7fkrel} to obtain a relationship between the investments $i_F, i_B$ and $i$ that need to hold to maintain the balanced growth path. In doing this, we use \eqref{eq:Enprice} to eliminate $p_e$ from \eqref{eq:Reg7etadot} and note that $\dot{h}=k_B$. Specifically, differentiating \eqref{eq:Reg7fkrel} results in a relationship between $i$ and $i_F$ that can be simplified to
% \begin{equation}
% \biggl [\lambda(A+m)-\eta\biggr ]i-\biggl [\lambda(\delta+m)-\eta\biggr ]i_F=\biggl \{(A-\delta)\eta+\delta\bigl [\lambda(A+m)-\eta\bigr ]\biggr\}k
% \label{eq:j0iiFrel}
% \end{equation}
% A second relationship  between the investments can be obtained by differentiating the energy market equilibrium condition \eqref{eq:Reg7EnEquil}, which leads to:
% \begin{equation}
% (F_0-af)i-aki_F-(H_0+bh)i_B=bk_B^2
% \label{eq:j0iiBiFrel}
% \end{equation}
% Equations  \eqref{eq:j0iiFrel} and \eqref{eq:j0iiBiFrel} can then be solved for $i_F$ and $i_B$ in terms of $i$. Substituting the results, along with the solution for $c$, into the budget constraint, we can then determine $i$ and thence $i_F$ and $i_B$.

% The upper boundary of this regime will occur where $h=(\bar{H}-H_0)/b$. We again require $k, \lambda$ and $f$ to be continuous across that boundary. To obtain the ``initial condition'' for the shadow price $\eta$, we use the transversality condition at $T_7$. Specifically, the value of $T_7$ is itself a choice variable since different initial value of $\eta$ will lead to different levels of investment in $h$ and hence different times at which $h$ attains its maximum value. Hence, the marginal effect of a change in $T_7$, namely the current value of the Hamiltonian at $T_7$ plus $\partial V/\partial t$, must equal zero.  But  $\partial V/\partial t$ is given by:
% \begin{equation}
% \frac{\partial}{\partial T_7}\biggl [\int_{T_7}^\infty e^{-\beta(\tau-T_7)}\frac{c(\tau)^{1-\gamma}}{1-\gamma} \,d\tau\biggr ]=-\frac{c(T_7)^{1-\gamma}}{1-\gamma} 
% \label{eq:MargTermValue}
% \end{equation}
% where $c$ is the optimal consumption path. The current value Hamiltonian will be given by \eqref{eq:Hamiltonian} evaluated at $T_7$. The budget constraint \eqref{eq:Budget} and the energy market equilibrium \eqref{eq:EnEquil} will both hold with equality. In this regime $j, \rho, n, i_R , q_R$ and $\sigma$ are all zero  while $q_B=\lambda=\varphi$. Hence, the Hamiltonian at $T_7$ will equal
% \begin{equation}
% \begin{split}
% \frac{c(T_7)^{1-\gamma}}{1-\gamma}+\lambda (i-\delta k+i_B-\delta k_B+i_F)+\eta k_B
% \end{split}
% \label{eq:HamT6}
% \end{equation}
% Thus, the transversality condition at $T_7$ will require
% \begin{equation}
% \eta(T_7)k_B=-\lambda(i-\delta k+i_B-\delta k_B+i_F)<0
% \label{eq:etaT6}
% \end{equation}
% Since learning by doing continues to increase $h$ beyond $T_6$, when explicit investment in $h$ ceases to be worthwhile, it is not surprising that the terminal value of its shadow price $\partial V/\partial h$ should be negative at $T_7$. From the first order condition \eqref{eq:FOCj} for $j$, throughout the preceding regime 6 we will have $\lambda>\eta \psi k_B$ so the lower boundary of regime 7 will occur where $\eta\psi k_B=\lambda>0$.

\subsection{Regime 5: Fully dynamic renewable energy regime}

Continuing to work backwards in time, regime 6 will have direct investment in improving the efficiency of renewable energy technology ($j>0$) in addition to productivity gains through learning by doing. The first order conditions for the choice of investments, \eqref{eq:FOCi}--\eqref{eq:FOCiF} and \eqref{eq:FOCj}, now lead to $\lambda=q=q_B=\varphi=\eta\psi k_B$. Once again, we find that the shadow price of energy is given by \eqref{eq:Enprice}. The additional co-state equation analogous to \eqref{eq:Reg7Costate} is
\begin{equation}
\dot{\lambda}=(\beta+\delta)\lambda-\eta(1+\psi j)+\lambda m-p_e(H_0+bh)
\label{eq:Reg6Costate1}
\end{equation}
In addition, differentiating $\lambda=\eta\psi k_B$ and using \eqref{eq:kBdot} and the co-state equation \eqref{eq:etadot} for $\eta$ we obtain
\begin{equation}
\dot{\lambda}=(\beta-\delta)\lambda+\psi\eta i_B-\psi p_ebk_B^2
\label{eq:Reg6Costate2}
\end{equation}

A major difference in this regime is that we can use equality of \eqref{eq:Reg6Costate1} and \eqref{eq:iiFPos} to solve for $j$  in terms of $p_e$ and the state and co-state variables:
\begin{equation}
\eta\psi j = (\delta+m)\lambda-\eta+p_e[ak-(H_0+bh)]
\label{eq:Reg6j}
\end{equation}
Similarly, from \eqref{eq:Reg6Costate2} and  \eqref{eq:iiFPos}, we can solve for $i_B$ in terms of $p_e$ and the state and co-state variables:
\begin{equation}
\eta\psi i_B=\delta\lambda+p_e(\psi bk_B^2-ak)
\label{eq:Reg6iB}
\end{equation}

As in regime 7, but now using $\dot{h}=k_B(1+\psi j)$, we can differentiate the energy market equilibrium condition \eqref{eq:Reg7EnEquil} to get an equation that can be solved for $i_F$ in terms of $i$ and the state and co-state variables (using the solutions \eqref{eq:Reg6j} and \eqref{eq:Reg6iB} for $i_B$ and $j$):
\begin{equation}
(F_0-af)i-aki_F=(H_0+bh)i_B+bk_B^2(1+\psi j)
\label{eq:Reg6iF}
\end{equation}
Finally, substituting the results \eqref{eq:Reg6j}, \eqref{eq:Reg6iB} and \eqref{eq:Reg6iF}, along with the solution for $c$, into the budget constraint, we can then determine $i$ and thence $i_F$.

The upper limit of this regime will occur where the solution to \eqref{eq:Reg6j} for $j$ equals zero. Before discussing the lower limit, we need to consider the fossil fuel regimes.

\subsection{Regime 1: Fossil fuels only}

We first show that fossil fuel capital will be fully utilized in regimes 1, 2 and 3:

\begin{Prop}
So long as $i, i_F, i_R>0$ we must also have $\rho=1$.
\label{Prop:rho=1}
\end{Prop}
\proof For $i, i_F>0$, \eqref{eq:FOCi} and \eqref{eq:FOCiF} imply $q=\lambda=\varphi$ and from \eqref{eq:qdot} and \eqref{eq:varphidot} we again conclude that $p_e$ is given by \eqref{eq:Enprice}. Now suppose also that $i_R>0$. Then \eqref{eq:FOCiR} implies $q_R=\lambda$ and hence also $\dot{q}_R=\dot{\lambda}$. But then \eqref{eq:qdot} and \eqref{eq:qRdot} would imply
\begin{equation}
\dot{\lambda}=(\beta+\delta)\lambda-\bigl [\sigma Q-\lambda g+p_e\bigr ]\rho\mu_R=(\beta+\delta)\lambda-A\lambda+p_e(F_0-af)
\label{eq:lamdotProp}
\end{equation}
In particular, using \eqref{eq:Enprice}, we can write
\begin{equation}
\bigl [\sigma Q-\lambda g+p_e\bigr ]\rho\mu_R=A\lambda-p_e(F_0-af)=\frac{\lambda\bigl [Aak+\delta(F_0-af)\bigr ]}{ak+F_0-af}>0
\label{eq:Prop_rho}
\end{equation}
and hence from \eqref{eq:FOCrho}, $\rho=1$. 

\endproof

It is useful to note next that the energy market equilibrium condition in this regime is given by
\begin{equation}
(F_0-af)k=\mu_Rk_R
\label{eq:Reg1EnEquil}
\end{equation}
 Differentiating this condition yields
\begin{equation}
(F_0-af)i-aki_F-\mu_Ri_R=0
\label{eq:Reg1InvestRel2}
\end{equation}

Next observe that since we also have $n>0$ in this regime, \eqref{eq:FOCn} will imply $\nu=\lambda$ while we already have $q=q_R=\varphi$. From the corresponding co-state equations, and using Proposition~\ref{Prop:rho=1}, we obtain, in addition to \eqref{eq:iiFPos} (which again yields $p_e$) the following two additional expressions for $\dot{\lambda}$:
\begin{equation}
\dot{\lambda}=(\beta+\delta)\lambda-\sigma Q\mu_R+\lambda g\mu_R-p_e\mu_R=\beta\lambda +\lambda\mu_Rk_R\frac{\partial g}{\partial N}
\label{eq:Reg1CoState}
\end{equation}
Equating each expression in \eqref{eq:Reg1CoState} to the second expression in \eqref{eq:iiFPos}, and substituting for $p_e$, we obtain two relationships between the capital variables $k, k_R, f, S$ and $N$ in this regime:
\begin{equation}
\begin{split}
\mu_Rk_R\frac{\partial g}{\partial N}(ak+F_0-af)&=-ak(A-\delta)
\\  \mu_R(ak+F_0-af)(\lambda g -\sigma Q)&=\lambda\biggl [(A-\delta)\mu_R-aAk-(F_0-af)\delta\biggr ]
\end{split}
\label{eq:Reg1CapRel1}
\end{equation}

Differentiating the second equation in \eqref{eq:Reg1CapRel1} with respect to time, noting that $\dot{Q}=\pi Q$ and using \eqref{eq:sigmadot} with $\rho=1$ to eliminate $\dot{\sigma}$, the final expression in \eqref{eq:iiFPos} to eliminate $\dot{\lambda}$, and the first equation in  \eqref{eq:Reg1CapRel1} to simplify the result, we obtain:
\begin{equation}
\begin{split}
&\biggl [\mu_R(\lambda g-\sigma Q)+\lambda A\biggr ]i-\biggl [\mu_R(\lambda g-\sigma Q)+\lambda \delta\biggr ]i_F-\frac{k(A-\delta)\lambda}{k_R}n
\\&=\delta k\biggl [\mu_R(\lambda g-\sigma Q)+\lambda A\biggr ]+\sigma Q\mu_Rk(A-\delta)-k(A-\delta)\pi\sigma Q\bigl (k_R\frac{\partial g}{\partial N}\bigr )^{-1}
\end{split}
\label{eq:Reg1InvestRel1}
\end{equation}
Also, using the energy market equilibrium condition \eqref{eq:Reg1EnEquil}, the first equation in \eqref{eq:Reg1CapRel1} can be simplified to
\begin{equation}
(F_0-af)\frac{\partial g}{\partial N}(ak+F_0-af)=-a(A-\delta)
\label{eq:Reg1CapRel2}
\end{equation}
where, in particular, the right hand side is now a constant. Differentiating \eqref{eq:Reg1CapRel2} and using \eqref{eq:Reg1CapRel2} and \eqref{eq:Reg1InvestRel2} to simplify yields
\begin{equation}
\begin{split}
&-2(F_0-af)\biggl (\frac{\partial g}{\partial N}\biggr )^2i_F+\mu_R\biggl (\frac{\partial g}{\partial N}\biggr )^2i_R-(A-\delta)\frac{\partial^2 g}{\partial N^2}n
\\ &=(F_0-af)\biggl (\frac{\partial g}{\partial N}\biggr )^2\delta k+(A-\delta)\frac{\partial^2 g}{\partial S\partial N}\mu_Rk_RQ
\end{split}
\label{eq:Reg1InvestRel3}
\end{equation}

The budget constraint, together with the first order condition \eqref{eq:FOCc}, will then give us a fourth equation. Specifically, equations  \eqref{eq:Reg1InvestRel2}, \eqref{eq:Reg1InvestRel1} and \eqref{eq:Reg1InvestRel3} can be solved for $i_F, i_R$ and $n$ as functions of $i$, which can then be substituted into the budget constraint to get an equation that can be solved for $i$. Having done so, we can then recover values for the remaining investments.

Having solved these equations for $i, i_F, i_R$ and $n$, the differential equations \eqref{eq:kdot}, \eqref{eq:EffGain}, \eqref{eq:kRdot}, \eqref{eq:Sdot} and \eqref{eq:Ndot} will determine the evolution of $k, f, k_R, N$ and $S$. The shadow price of output, $\lambda$, will evolve according to any equation in \eqref{eq:Reg1CoState}, while \eqref{eq:sigmadot} with $\rho=1$ will determine the evolution of $\sigma$. Note that throughout regime 1 we will have $k_B=H=0=\dot{k}_B=\dot{H}$ and $\eta$ constant in present value terms.

We shall guess initial values for the co-state variables $\sigma$ and $\nu$ at $t=0$. Ultimately, however, these will be determined to match terminal conditions discussed in more detail below.

We also need to guess an initial value $\eta_0$ for $\eta$. As noted already, $\eta$ will satisfy a terminal condition at $T_7$, and ultimately this terminal condition will determine the initial value $\eta_0$. However, for a given guess for $\eta_0$, the upper boundary of regime 1, $T_1$, will be determined where $q=q_B=\lambda$ and $\dot{q}_B=\dot{q}=\dot{\lambda}$, that is,
\begin{equation}
\eta_0=\lambda(A+m) - p_e(H_0+F_0-af)
\label{eq:T1condition}
\end{equation}


\subsection{Regime 2: Fossil fuels and renewables both used}

With $i, i_F, i_R, n$ and $i_B$ positive Proposition~\ref{Prop:rho=1} implies $\rho=1$. Then \eqref{eq:FOCi}, \eqref{eq:FOCiR}, \eqref{eq:FOCiB},  \eqref{eq:FOCiF} and \eqref{eq:FOCn} imply $\lambda=q=\varphi=q_R=q_B=\nu$, so using \eqref{eq:qdot}, \eqref{eq:qRdot}, \eqref{eq:qBdot},  \eqref{eq:nudot} and \eqref{eq:varphidot} we again obtain \eqref{eq:iiFPos}, and hence expression \eqref{eq:Enprice} for $p_e$, and the additional equations \eqref{eq:Reg1CoState} that also were applicable in the fossil fuel regime 1, but we now also add equation \eqref{eq:Reg7Costate} that applies in regime 7 when we also have $i_B>0$ and $j=0$. The energy market equilibrium condition in this region is given by:
\begin{equation}
(F_0-af)k=\mu_Rk_R + (H_0+hb)k_B
\label{eq:Reg2EnEquil}
\end{equation}
and will provide an additional equation linking $k, k_R, k_B$ and $f$.

As in regime 7,  equation \eqref{eq:Reg7Costate} can be combined with \eqref{eq:iiFPos} and \eqref{eq:Enprice} to yield \eqref{eq:Reg7fkrel}, which can then be differentiated to yield equation \eqref{eq:j0iiFrel} linking $i$ and $i_F$.

Similarly, the co-state equations relating to fossil fuels \eqref{eq:Reg1CoState} again yield the pair of equations \eqref{eq:Reg1CapRel1}  once we have substituted the solution \eqref{eq:Enprice} for $p_e$. Differentiating the second of these equations we again obtain \eqref{eq:Reg1InvestRel1}. However, we do not obtain \eqref{eq:Reg1InvestRel3} from the first equation since we used the energy market equilibrium condition applicable in regime 1 to obtain that equation rather than the new condition \eqref{eq:Reg2EnEquil}. Differentiating the first equation in \eqref{eq:Reg1CapRel1} we now obtain a new equation linking investments:
\begin{equation}
\begin{split}
\frac{\partial g}{\partial N}\biggl [\frac{\partial g}{\partial N}+ & \frac{A-\delta}{\mu_Rk_R}\biggr ]i-\biggl (\frac{\partial g}{\partial N}\biggr )^2i_F+\biggl (\frac{\partial g}{\partial N}\biggr )^2\frac{ak+F_0-af}{ak_R}i_R-\frac{\partial^2 g}{\partial N^2}\frac{k(A-\delta)}{\mu_Rk_R}n
\\&=\frac{\partial^2 g}{\partial S\partial N}Qk(A-\delta)+\biggl (\frac{\partial g}{\partial N}\biggr )^2\delta k
\end{split}
\label{eq:Reg2InvestRel1}
\end{equation}
A fourth equation linking the investments can be found by differentiating the energy market equilibrium condition \eqref{eq:Reg2EnEquil}. This yields
\begin{equation}
(F_0-af)i-aki_F-\mu_Ri_R-(H_0+hb)i_B=bk_B^2
\label{eq:Reg2InvestRel2}
\end{equation}
Equations \eqref{eq:j0iiFrel}, \eqref{eq:Reg1InvestRel1}, \eqref{eq:Reg2InvestRel1} and \eqref{eq:Reg2InvestRel2} then give four equations in the five investments $i, i_F, i_R, n$ and $i_B$. Once again, the budget constraint provides the final equation.

Given solutions for the investments, we then obtain five differential equations for the evolution of the state variables. Population will once again evolve exogenously at the rate $\pi$, so $\dot{Q}=\pi Q$. In this regime, $\lambda$ and $\sigma$ will continue to be given by the same equations as in regime 1. Now, however, the value of the co-state variable $\eta$ will also evolve:
\begin{equation}
\dot{\eta}=\beta\eta-p_ebk_B
\label{eq:etadotReg2}
\end{equation}
The upper boundary of regime 2 likely will occur at $T_2$ where $k_B=\lambda/\eta\psi$ and $j$ becomes positive. We would then enter a regime where all six investments are positive.

\subsection{Regime 3: All six investments positive}

If all six investments are positive, equations \eqref{eq:FOCi}--\eqref{eq:FOCj} would imply $\lambda=q=q_R=q_B=\varphi=\nu=\eta\psi k_B$. In turn,  $\dot{\lambda}=\dot{q}=\dot{q}_R=\dot{q}_B=\dot{\varphi}=\dot{\nu}=\psi(\dot{\eta}k_B+\eta\dot{k}_B)$ and, noting that $\rho=1$, equations \eqref{eq:qdot}, \eqref{eq:qRdot}, \eqref{eq:qBdot}, \eqref{eq:varphidot}, \eqref{eq:nudot} and \eqref{eq:etadot} yield a set of co-state variable differential equations that combine the equations for regimes 1 and 6. In particular, $j$ is again given by \eqref{eq:Reg6j} and $i_B$ by \eqref{eq:Reg6iB}. However, the energy market equilibrium condition differs from regime 6, so we no longer obtain \eqref{eq:Reg6iF} for $i_F$. Indeed, although the energy market equilibrium condition is once again given by \eqref{eq:Reg2EnEquil}, now $\dot{h}=k_B(1+\psi j)$. Upon differentiation, we now obtain
\begin{equation}
(F_0-af)i-aki_F-\mu_Ri_R=bk_B^2(1+\psi j)+(H_0+hb)i_B
\label{eq:Reg3InvestRel1}
\end{equation}
As in regime 2, the different expressions for $\dot{\lambda}$ again yield  \eqref{eq:Reg1InvestRel1} and \eqref{eq:Reg2InvestRel1}. We thus obtain three equations in the four investments $i, i_F, i_R$ and $n$. The budget constraint again supplies the final equation. The upper boundary of regime 3 occurs at $T_3$ when $i_R=0$.

\subsection{Regime 4: Investment in $k_R$ is zero, but $n$ remains positive}

When $i_R$ falls to zero, we no longer have $q_R=\lambda$. However, \eqref{eq:FOCiB}--\eqref{eq:FOCj} still imply $\lambda=q_B=\nu=\eta\psi k_B$. The following Proposition also shows that we retain the result $\rho=1$ even though $i_R=0$:
\begin{Prop}
If $i, i_F, n, k_R>0$, we must also have $\rho=1$.
\label{Prop:rhoReg4}
\end{Prop}
\proof For $i, i_F>0$, \eqref{eq:FOCi} and \eqref{eq:FOCiF} imply $q=\lambda=\varphi$ and from \eqref{eq:qdot} and \eqref{eq:varphidot} we again conclude that $p_e$ is given by \eqref{eq:Enprice}. Now suppose also that $n>0$. Then \eqref{eq:FOCn} implies $\lambda=\nu$ and from \eqref{eq:varphidot} and \eqref{eq:nudot} we obtain $\lambda\rho\mu_Rk_R\partial g/\partial N=-p_eak<0$. Hence, we must have $\rho>0$ and $\dot{\lambda}/\lambda=\dot{\varphi}/\varphi=\dot{\nu}/\nu<\beta$. In addition, $n, \rho>0$ and \eqref{eq:qRdot} imply that we cannot have $\dot{q}_R=0=q_R$ in regime 4. Now observe that \eqref{eq:FOCiR} implies $\lambda\ge q_R$ throughout regime 4, with equality at the beginning of the regime. Hence, $q_R$ must change  in regime 4 at some rate that is bounded above by $\beta$. But then \eqref{eq:qRdot} implies $\rho\mu_R\bigl (\sigma Q-\lambda g+p_e\bigr)>\delta q_R>0$, so we must have $\mu_R\bigl (\sigma Q-\lambda g+p_e\bigr)>0$. But then \eqref{eq:FOCrho} and $k_R>0$ imply $\theta_{U}>0$ and hence $\rho=1$.
\endproof

Now the first order conditions imply $\lambda=q=q_B=\varphi=\nu=\eta\psi k_B$ and we again obtain expression \eqref{eq:Enprice} for $p_e$, with $j$ again given by \eqref{eq:Reg6j} and $i_B$ by \eqref{eq:Reg6iB}. Although the energy market equilibrium condition is again  given by \eqref{eq:Reg2EnEquil}, now $\dot{k}_R=-\delta k_R$ and upon differentiation we obtain
\begin{equation}
(F_0-af)i-aki_F=bk_B^2(1+\psi j)+(H_0+hb)i_B
\label{eq:Reg4InvestRel1}
\end{equation}
Although the second equation in \eqref{eq:Reg1CapRel1} no longer holds since we now do not have $q_R=\lambda$, the first equation remains valid since we still have $\nu=\lambda$. When differentiating  the first equation in \eqref{eq:Reg1CapRel1}, however, we now have $\dot{k}_R=-\delta k_R$ and hence we find
\begin{equation}
\begin{split}
&a\biggl [\mu_Rk_R\frac{\partial g}{\partial N}+A-\delta\biggr ]i-a\mu_Rk_R\frac{\partial g}{\partial N}i_F+\mu_Rk_R\frac{\partial^2 g}{\partial N^2}(ak+F_0-af)n
\\&=\frac{\partial^2 g}{\partial S\partial N}Qak(A-\delta)\bigl (\frac{\partial g}{\partial N}\bigr )^{-1}+\mu_Rk_R\frac{\partial g}{\partial N}a\delta k
\end{split}
\label{eq:Reg4InvestRel2}
\end{equation}
Again the budget constraint will supply the final equation needed to solve for $i, i_F$ and $n$. The upper limit of regime 4 will occur at $T_4$ when the solution for $n$ in regime 4 equals zero.

\subsection{Regime 5: Fossil fuel use phases out}

While we have argued that $\rho=1$ while $n>0$, we have not shown that $\rho=0$ as soon as $n$ falls to zero. Once $n=0$, $N$ will remain constant but continued use of fossil fuel will still cause $S$ to increase. With $S$ increasing and $N$ constant, however, the marginal cost $g(S,N)$ of fossil fuel input will rise rapidly until fossil fuels become too expensive and are abandoned. Since this is likely to occur well before $S$ reaches $\bar{S}$, fossil fuel resources will be abandoned not because they are exhausted but simply because they become uneconomic relative to the alternatives.

With $n=0$ we lose the result from \eqref{eq:FOCn} that $\nu=\lambda$ and hence even the first equation in \eqref{eq:Reg1CapRel1} no longer holds. Of course, we also lose one endogenous variable, since in this regime $n=0$. The equations to be solved for $i, i_F, i_B$ and $j$ are exactly the same as in the following regime 6, namely \eqref{eq:Reg6j}, \eqref{eq:Reg6iB}, \eqref{eq:Reg6iF} and the budget constraint. The only difference from regime 6 is that $S, \sigma$ and $\nu$ will continue to evolve in regime 5 but not regime 6. In addition, we retain $\dot{k}_R=-\delta k_R$ as in regime 4. The following proposition shows that we retain the result from regime 4 that $\rho=1$.
\begin{Prop}
If $i, i_F, i_B,  j>0$, $n=0$ and $k_R>0$ then $\rho=1$.
\label{Prop:rhoReg5}
\end{Prop}
\proof From $i, i_F>0$ and equations \eqref{eq:FOCi} and \eqref{eq:FOCiF} we again obtain $q=\lambda=\varphi$. Then from \eqref{eq:qdot} and \eqref{eq:Enprice} we conclude that $\dot{\lambda}=\beta\lambda-\alpha(A-\delta)ak$. In particular, as in regime 4, $\dot{\lambda}/\lambda<\beta$. Then since $\nu=\lambda$ at $T_4$, but \eqref{eq:FOCn} requires $\lambda\ge\nu$ in regime 5, we must have $\dot{\nu}/\nu<\beta$ and hence from \eqref{eq:nudot} $\rho>0$. But then \eqref{eq:FOCrho} implies $\theta_L=0$ and $\theta_U=k_R\mu_R(\sigma Q-\lambda g+p_e)$. Now suppose, contrary to the claim, that $\rho<1$. Then we must have $\theta_U=0=\sigma Q-\lambda g+p_e$ with $p_e=\lambda(A-\delta)/(ak+F_0-af)$. Hence, $\lambda(A-\delta)=(\lambda g-\sigma Q)(ak+F_0-af)$. Differentiating this expression and using $\dot{\lambda}=\beta\lambda-\alpha(A-\delta)ak$, \eqref{eq:sigmadot}, $\dot{Q}=\pi Q$ and $n=0$ we obtain $\sigma Q(ak+F_0-af)[\pi-(A-\delta)ak]=(\lambda g-\sigma Q)(ai-\delta ak-ai_F)$. However, for $i, i_F, i_B,  j>0$ we already have $i, i_F, i_B$ and $j$ determined by \eqref{eq:Reg6j}, \eqref{eq:Reg6iB}, \eqref{eq:Reg6iF} and the budget constraint and these solutions are inconsistent with the equation for $i$ and $i_F$ just derived. Hence, we must have $\rho=1$.

\endproof

Since $\rho=1$ we conclude that $\sigma$ and $\nu$ will evolve according to $\dot{\sigma}=\beta\sigma+\lambda\mu_Rk_R\partial g/\partial S$ and $\dot{\nu}=\beta\nu+\lambda\mu_Rk_R\partial g/\partial N$. At the upper boundary of regime 5, $\sigma=\nu=0$. At that time, the shadow price of energy has to satisfy $p_e=\lambda g(S,N)$, which using \eqref{eq:Enprice}, implies
\begin{equation}
g(S,N)=\frac{A-\delta}{ak+F_0-af} 
\label{eq:peatT5}
\end{equation}
Equation \eqref{eq:peatT5} provides a condition that can be used to determine $T_5$. The requirement that $\sigma=\nu=0$ at $T_5$ then effectively provides two conditions to determine the initial values of the co-state variables $\sigma$ and $\nu$ at $t=0$.

Also observe from \eqref{eq:FOCiR} that $\omega_R=\lambda-q_R\ge 0$ and this must remain valid for $t\ge T_5$. Also, since $\rho=0$ for $t>T_5$, if $q_R\neq 0$ at $T_5$, \eqref{eq:qRdot} implies $q_R$ would increase at the rate $\beta+\delta$ for $t>T_5$. On the other hand, since $i_F>0$ for $t\ge T_5$, \eqref{eq:FOCiF} will imply $\varphi=\lambda$. Then \eqref{eq:varphidot} would imply that $\lambda$ increases at a rate lower than $\beta$. Hence, we must have $q_R=0, \forall t>T_5$, so the shadow value of capital that uses fossil fuel as an energy source must be zero once such capital is no longer in use.

\subsection{Calibration}

In order to quantitatively evaluate different policy scenarios, we first need to calibrate the theoretical model. This involves assigning numerical values to certain parameters in a way that make the model consistent with observations from the actual world economy. By definition, we start the economy with $S=N=H=k_B=0$ and with $Q=Q_{0}$. For convenience, we take the current population $Q_{0}=1$ and effectively measure future population as multiples of the current level. We will assume that the population growth rate is 1\%.\footnote{This is consistent with a simple extrapolation of recent world growth rates reported by the Food And Agriculture Organization of the United Nations, \textsf{http://faostat.fao.org/site/550/default.aspx}}

In line with standard assumptions made to calibrate growth models, we assume a time discount factor $\beta =0.05$. From previous analyses of macroeconomic and financial data, we would expect the coefficient of relative risk aversion $\gamma$ to lie between 1 and 10, but there is no strong consensus on what the value should be. As we explain in more detail below, we will allow $\gamma$ to adjust to ensure we match the initial share of consumption in GDP.

To calibrate values for the initial production, capital stocks and energy quantities we used data from the \textit{Energy Information Administration} (EIA),\footnote{International data is available at \textsf{http://www.eia.doe.gov/emeu/international/contents.html}} the \textit{Survey of Energy Resources 2007} produced by the \textit{World Energy Council},\footnote{This is available at \textsf{http://www.worldenergy.org/publications/survey\_of\_energy\_resources\_2007/default.asp} The data are estimates as of the end of 2005.} and \textit{The GTAP 7 Data Base} produced by the \textit{Center for Global Trade Analysis} in the Department of Agricultural Economics, Purdue University.\footnote{Information on this can be found at \textsf{https://www.gtap.agecon.purdue.edu/databases/v7/default.asp} The GTAP 7 data base pertains to data for 2004.} The last mentioned data source is useful for our purposes because it provides a consistent set of international accounts that also take account of energy flows.

One of the first issues we need to address is that national accounts include government spending in GDP, which does not appear in the model.\footnote{Note that in the GTAP data base, aggregate world exports equal aggregate world imports so world GDP equals consumption plus investment plus government expenditure.} We therefore subtracted government spending from the GDP measures before calibrating the remaining variables. Conceptually, this would be correct if the utility obtained from government spending were additively separable from the utility obtained from private consumption and government spending was financed by lump sum taxes. In practice, neither of these assumptions is valid and government activity (apart from energy taxes or subsidies, which will be considered explicitly later) would affect the equilibrium of the model.

After excluding government, the investment share of private sector expenditure is 0.2569. Effectively defining units so that aggregate output is 1, we therefore identify 0.2569 as the sum $i+i_F+i_R+n$ at $t=0$. We would expect most of this to be investment in capital used to produce final output rather than end-use energy efficiency improvements or improvements in fossil fuel mining and conversion activities.

Converting the GTAP data base estimates of the total capital stock capital stock to units of GDP, we obtain the initial condition $k(0)+k_R(0)=3.2802$. The GTAP data does not allocate the capital stock to different sectors, but it does give new capital purchases by sector. Identifying mining of coal, oil, and natural gas, refining, electricity production from coal, oil and natural gas, and gas distribution as the fossil fuel mining and conversion sectors, we obtain that 6.91\%, or 0.2267 of the capital would be $k_R(0)$ while $k(0)=3.0535$.

Given that we have chosen units so the $R=1$, this also implies that the energy efficiency parameter $\epsilon=1/k_R(0)=0.2772$. Similarly, if we choose GDP units so that output equals 1, the parameter $A$ would equal the ratio of output to capital, that is, $A$ also is 0.2772. We also use the GTAP depreciation rate on capital of 4\%.

From the budget constraint, the difference between total output and the sum of the investments, namely 0.7431 would equal consumption plus the current costs $gR$ of supplying fossil fuels. We separated these two components using sectoral data from the GTAP data base. Specifically, we classified ``energy expenditure'' as combined spending on the primary fuels coal, oil and natural gas, and the energy commodity transformation sectors of refining, electricity generation and natural gas distribution. In effect, we are associating $gR$ with the costs involved in mining fossil fuels and turning them into commodities capable of providing energy services to productive capital. The current cost of fossil energy was then set equal to the expenditure on these sectors that was classified as consumption rather than investment. This produced a value for $gR=0.0565$. Observe that, assuming $R=1$, this value of $g$ implies fossil fuels yield positive net output at $t=0$, that is, $A-\delta-g/\epsilon=0.0335>0$ but an increase of $g$ to just 0.0658 would erase this marginal surplus.

Subtracting the initial value for $gR$ from 0.7431 we obtain the initial value of $c(0)=0.6867$. As noted above, the normal method of solving the optimal control problem would involve specifying values for the parameters and the state variables and then solving for values of the co-state variables that allow us to hit required terminal values. The value for $c(0)$ would then follow from the first order condition $\lambda(0)=c(0)^{-\gamma}$. To obtain a particular value for $c(0)$ we need to free up an additional parameter. As already noted above, we will introduce $c(0)$ as a new target and adjust the value of $\gamma$ as $\lambda(0)$ changes to ensure that  $\lambda(0)=c(0)^{-\gamma}$ always remains valid.

After we set the initial values of $S$ and $N$ to zero, the initial value for $gR$ also would imply 
\begin{equation}
\frac{0.0565}{R}=\alpha _{0}+\frac{\alpha _{1}}{\bar{S}-\alpha _{2}/\alpha_{3}}
\label{eq:Init_g}
\end{equation}
We can choose energy units so that the initial value of $R=1$ by definition. However, we still need to know the worldwide annual production of fossil fuels in order to calibrate other energy terms in the same units. The EIA web site gives world wide production of oil in 2005 of 175.896 quads (where one quad equals $10^{15}$ BTU), of natural gas 100.141 quads and of coal 123.03 quads. Summing these gives a total of 392.637 quads, which we will take as our measure of one unit of ``energy services''.

To obtain an estimate of total fossil fuel resources $\bar{S}$ in the same units, we used data from the World Energy Council and the US Geological Survey (USGS). The millions of tonnes of coal, millions of barrels of oil, extra heavy oil, natural bitumen and oil shale and trillions of cubic feet of conventional and unconventional natural gas were converted to quads using conversion factors available at the EIA. The result is 21.780 quintillion BTU of coal, 20.369 quintillion BTU of conventional and unconventional oil and . These resources are nevertheless relatively small compared to estimates of the volume of methane hydrates that may be available. Although experiments have been conducted to test methods of exploiting methane hydrates, a commercially viable process is yet to be demonstrated. Partly as a result, resource estimates vary widely. According to the National Energy Technology Laboratory (NETL),\footnote{\textsf{http://www.netl.doe.gov/technologies/oil-gas/FutureSupply/MethaneHydrates/about-hydrates/estimates.htm}} the United States Geological Survey (USGS) has estimated potential resources of about 200,000 trillion cubic feet in the United States alone. According to Timothy Collett of the USGS,\footnote{\textsf{http://www.netl.doe.gov/kmd/cds/disk10/collett.pdf}} current estimates of the worldwide resource in place are about 700,000 trillion cubic feet of methane. Using the latter figure, this would be equivalent to 719.6 quintillion BTU. Adding this to the previous total of oil, natural gas and coal resources yields a value for $\bar{S}=834.8$ quintillion BTU or around 2126.1769 in terms of the energy units defined so that $R=1$.

We still need to specify values for the $\alpha_i$ parameters in the $g$ function. Equation \eqref{eq:Init_g} with $R\equiv 1$ will give us one equation in four unknowns. Noting that we can interpret $\bar{S}-\alpha_2/\alpha_3$ as the initial level of fossil fuel extraction $S$ at which marginal costs of extraction $g(S,0)$ would become unbounded, we associate  $\bar{S}-\alpha_2/\alpha_3$ with current proved and connected reserves of fossil fuel.\footnote{Note that current official reserves are not the relevant measure since many of these are not connected and thus are unavailable for production without further investment, denoted $n$ in the model.} A recent report from Cambridge Energy Research Associates (CERA, 2009),\footnote{``The Future of Global Oil Supply: Understanding the Building Blocks,'' Special Report by Peter Jackson, Senior Director, IHS Cambridge Energy Research Associates, Cambridge, MA.} for example, gives weighted average decline rates for oil production from existing fields of around 4.5\% per year. They also note that this figure is dominated by a small number of ``giant'' fields and that, ``the average decline rate for fields that were actually in the decline phase was 7.5\%, but this number falls to 6.1\% when the numbers are production weighted.'' As an approximation, we shall use 6\% as a decline rate for oil fields. Using United States natural gas production and reserve figures as a guide, we find that natural gas decline rates are closer to 8\% per year. The United States data coal mine decline rates approximate 6\% per year. In accordance with these figures, we assume the ratio of fossil fuel production to proved and connected reserves equals the share weighted average of these figures, namely $(175.948\ast 0.06+100.141\ast 0.08+116.6\ast 0.06)/392.689=0.0651$. Thus, in terms of the energy units defined so that $R=1$, the initial target value of $\bar{S} -\alpha_2/\alpha_3$ would equal 1/0.0651=15.361. Using the previously calculated value for $\bar{S}$, this leads to $\alpha_2/\alpha_3=2110.538$.

We can obtain two more equations by examining the investment in fossil fuel production at $t=0$. Using GTAP data on capital shares by sector, we estimate that around 8.4\% of annual investment occurs in the oil, natural gas, coal, refining, electricity, and gas distribution sectors. We noted above that in the GTAP data, total investment $i+n=0.2569$, implying that $n\approx 0.0215$ in private sector output units. We assume that this level of investment at $t=0$ is sufficient to replace mined resources and allow for growth in total annual fossil fuel production equivalent to the average annual growth over 2004-08 of around 2.35\%.\footnote{These calculations are again based on data from the EIA.} Specifically, with $\alpha_2/\alpha_3=2110.538$, we assume that the investment $n=0.0215$ increases reserves by the amount mined $R=1$ plus 2.35\% of 15.361, that is, $\alpha_2/(\alpha_3+0.0215)=2109.195$, which implies $\alpha_3\approx 33.261$. The previously calculated value for $\alpha_2/\alpha_3$ then implies $\alpha_2 \approx 70198.52$. Next, we observe that the partial derivatives of $g$ depend on $\alpha_1, \alpha_2$ and $\alpha_3$ but not on $\alpha_0$. Using the previously calculated values for $c(0)$ and $k_R(0)$ and the guessed values for $\lambda(0)$ and $\sigma(0)$ we choose $\alpha_1$ so that equation \eqref{eq:nReg1} solves for $n=0.0215$. Specifically, using the expressions \eqref{eq:PartialgPartialN}, \eqref{eq:Partial2gPartialN2} and \eqref{eq:Partial2gPartialSN} for the partial derivatives of $g$, \eqref{eq:nReg1} becomes a linear equation in $\alpha_1$:
\begin{equation}
\begin{split}
\frac{\alpha_1\alpha_2}{(\bar{S}\alpha_3-\alpha_2)^2}&\biggl [2n(0)\biggl (\frac{k_R(0)\bar{S}}{\bar{S}\alpha_3-\alpha_2}+1\biggr )+k_R(0)\biggl (\delta-A+\frac{\sigma(0)}{\lambda(0)\epsilon}+\frac{g(0)}{\epsilon}\biggr )+c(0)-\frac{2\alpha_3k_R(0)^2}{\epsilon(\bar{S}\alpha_3-\alpha_2)}\biggr ]
\\ & =-\frac{\sigma(0)\pi}{\lambda(0)}
\end{split}
\label{eq:alpha1eqn}
\end{equation}
Finally, using the calibrated value of 0.0565 for $g$ we can then solve for $\alpha_0=g-\alpha_1/(\bar{S}-\alpha_2/\alpha_3)$.

Turning next to the learning curve \eqref{eq:RenewCost}, the literature
provides a range of estimates for $\alpha $. An online calculator provided
by NASA\footnote{Available at \textsf{http://cost.jsc.nasa.gov/learn.html}} gives a range of
learning percentages between 5 and 20\% depending on the industry. A
learning percentage of $x$, which corresponds to a value of $\alpha
=-ln(1-x)/ln(2)$, has the interpretation that a doubling of the experience
measure will lead to a cost reduction of $x$\%. Thus, $x=0.2$ is equivalent
to $\alpha =0.322$ while $x=.05$ corresponds to $\alpha =0.074$. In a study
of wind turbines, Coulomb and Neuhoff (2006)\footnote{Louis Coulomb and Karsten Neuhoff, \textquotedblleft Learning Curves and
Changing Product Attributes: the Case of Wind Turbines\textquotedblright ,
University of Cambridge: Electricity Policy Research Group, Working Paper
EPRG0601.} found values of $\alpha $ of 0.158 and 0.197. In a 1998 paper, Gr\"{u}bler and Messner\footnote{Arnulf Gr\"{u}bler and Sabine Messner, \textquotedblleft Technological
change and the timing of mitigation measures\textquotedblright , \textit{Energy Economics} 20, 1998, 495--512} found a value for $\alpha =.36$ using
data on solar panels. In a 2008 paper in \textit{The Energy Journal}, van
Bentham et. al.\footnote{\textquotedblleft Learning-by-doing and the optimal solar policy in
California,\textquotedblright\ Arthur van Benthem, Kenneth Gillingham and
James Sweeney, 29(3) 2008, 131-152} report several studies finding a
learning percentage of around 20\% ($\alpha =0.322$) for solar panels. For
our base case, we will take $\alpha =0.25$.

The other parameter affecting the incentive to invest in renewable energy sources is the initial value $\Gamma _{1}^{-\alpha }$ of the cost of renewable energy capital relative to fossil fuel capital. Using a document available from the Energy Information Administration (EIA) \footnote{\textit{Assumptions to the Annual Energy Outlook, 2010} available at \textsf{http://www.eia.doe.gov/oiaf/aeo/electricity\underline{ }generation.html}} the levelized capital cost of new onshore wind capacity is about six times the cost of combined cycle gas turbines (CCGT), while offshore wind is around seven times as expensive, solar thermal about ten times as expensive and solar photovoltaic more than seventeen times as expensive. On the other hand, geothermal and biomass capacity is only about four times as expensive, and nuclear and hydro about five times as expensive as CCGT. As a rough approximation, we will assume that the initial value of $\mu=\Gamma _{1}^{-\alpha }$ is around 5. We also assume that, in the long run, the renewable technologies can experience a five-fold reduction in costs, so their long run capital cost would be about 20\% above the current capital cost of fossil fuel technologies.

To fix the O\&M cost for renewables, $m$, we note that the same EIA document gives a fixed O\&M cost of onshore
wind that is around one-fifth the corresponding fixed plus variable (including fuel cost) O\&M for CCGT. The corresponding ratio is around one-half for offshore wind, one-tenth for solar photovoltaic, around one-half for nuclear and geothermal and one-fifth for hydro. As an approximation, we assume that $m$ equals 0.2 times the initial value of $g$, that is, 0.0113. Observe that for these values of $m,\delta, \Gamma_1$ and the previously set value for $A$, $A\Gamma_1^\alpha-\delta-m=0.0042>0$ as we assumed.

Finally, we need to specify a value for $\psi $, the relative effectivenessof direct investment in research versus learning by doing in accumulatingknowledge about new energy technologies. Klaassen et. al. (2005)\footnote{Klaassen, Ger, Asami Miketa, Katarina Larsen and Thomas Sundqvist,\textquotedblleft The impact of R\&D on innovation for wind energy inDenmark, Germany and the United Kingdom,\textquotedblright\ \textit{Ecological Economics}, 54 (2005) 227--240} estimated a model that allowedfor both learning-by-doing and direct R\&D. Although they assume the capitalcost is multiplicative in total R\&D and cumulative capacity, while we assume the \textit{change} in knowledge is multiplicative in new R\&D and cumulativecapacity, we can take their parameter estimates as a guide. They find directR\&D is roughly twice as productive for reducing costs as islearning-by-doing.\footnote{Of course, the learning-by-doing has the advantage that it directlycontributes to output at the same time it is adding to knowledge.} Consequently, we assume that $\psi =2$.

\newpage

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